It sounds like you're running into a form of copy protection from your cable provider called SOC (Selectable Output Control). In a nutshell, because the cable box already has an HDMI connection, it shuts off the video signal coming through other connections, such as the component cables connecting running to your Slingbox.
There are two workarounds for this:
1) Disconnect the HDMI cable entirely, and run a second set of component cables OUT from the Slingbox to the TV.
2) If you want/need to keep using HDMI for your TV, we recommend the use of an HDMI to component adapter. Here is some information on this:
Hope this helps!